I've always been under the impression that the dish washing soap was added to poisons and fertilizers to help the chemistry hold to the plant. That without the soap the majority of the chemicals would run off the leafs with the water.
Does the dish soap actually have the ability to kill insects?
I doesn't seem like it would. What aspect of the soap kills? I've read and experimented with several home remedies for pest, all of which contained dish soap, but they always also contained some sort of agent that had a "killing" aspect to it.. such as tobacco or something. Caution with tobacco: there is a caterpillars that thrives on tobacco and home pesticide remedies often attract them.
On the lady bugs... If you live in Florida, scare all the lizards away before you release the little ladies!! I bought a bag of them once and created a lizard feast. I felt bad for the poor lady bugs <grin>.